Consider the unit Eucleidian sphere \( \mathbb{S}^{m-1}=\left \{ x \in \mathbb{R}^m :\left \| x \right \|_2=1 \right \} \) in \( \mathbb{R}^m \). We will define "distance" \( d(x, y) \) of two points \( x, y \in \mathbb{S}^{m-1} \) to be the convex angle \( xO y \) that is defined by the origin and the \( x, y \) points.

a) Show that if \( d\left ( x, y \right )=\theta \) then \( \displaystyle \left \| x-y \right \|_2=2\sin \frac{\theta }{2} \) holds.

b) Is \( d \) a metric in \( \mathbb{S}^{m-1} \)?

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Imagination is much more important than knowledge.

Let \(\displaystyle{x\,,y\in S^{m-1}}\) such that \(\displaystyle{d(x,y)=\theta}\).

Consider the perpendicular to \(\displaystyle{xy}\) - line from the origin. Then,

\(\displaystyle{\sin\,\dfrac{\theta}{2}=\dfrac{||x-y||_{2}}{2}\iff ||x-y||_{2}=2\,\sin\,\dfrac{\theta}{2}}\).

Yes, the function \(\displaystyle{d}\) is a metric in \(\displaystyle{S^{m-1}}\) and with this metric,

the triplet \(\displaystyle{\left(S^{m-1}\,,d\,,\sigma\right)}\), where,

\(\displaystyle{\sigma(A)=\dfrac{|\left\{s\,x\in \mathbb{R}^m\,,0\leq s\leq 1\right\}|}{|B_{2}^m|}\,,\forall\,A\in\mathbb{B}(S^{m-1})}\),

becomes a metric probability space.